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Wave function of the Lieb Liniger gas

In this section we will describe the construction of the wave function whis is used to solve the Lieb-Liniger equation in presence of an external confinement (refer to Sec. 3.2.5). The aim is to obtain a wave function suitable for the description of the gas in a wide range of the density, starting from the Tonks-Girardeau and up to the Gross-Pitaevskii regimes. The important point is that TG wave function always has nodes, although in GP regime the nodes are absent.

At the distances $\vert z\vert>0$ the interaction potential is absent and the solutions are are simple sinus and cosine functions. We want to choose a solution which goes to one at the matching distance $R_{m}$, which is treated as a variational parameter, and is a solution of a two-body problem (1.65) at a smaller distances. Also we want to have a symmetry in sign reversing. The solution that satisfies those conditions is

$\displaystyle f_2(z)
=\left\{
{\begin{array}{ll}
\displaystyle 1, & z < -R_{m}\...
...m}), & 0 \le z < R_{m}\\
\displaystyle 1, & R_{m}\le z\\
\end{array}}
\right.$     (2.51)

At the matching points $\pm R_{m}$ the derivative is automatically equal to zero and the function matches smoothly to a constant. The phase $\Delta(k) = kR_{m}$ is related to the scattering length $a_{1D}$ by the boundary condition (1.70), which we will write as2.2

$\displaystyle ka_{1D}\,\mathop{\rm tg}\nolimits kR_{m}= 1$     (2.52)

Once $\k$ is obtained by solving numerically this equation the drift force contribution (2.39) can be calculated from formula

$\displaystyle {{\cal F}_2}(z)
=\left\{
{\begin{array}{ll}
\displaystyle -k \mat...
...ert < R_{m}\\
\displaystyle 0, & \vert z\vert\ge R_{m}\\
\end{array}}
\right.$     (2.53)

The energy contribution (refEE) is then described by

$\displaystyle {{\cal E}_2}(z) =
\left\{
{\begin{array}{ll}
\displaystyle k^2(1+...
...ert < R_{m}\\
\displaystyle 0, & \vert z\vert\ge R_{m}\\
\end{array}}
\right.$     (2.54)


next up previous contents
Next: Phonon trial wave function Up: One-dimensional wave functions Previous: Hard-rod wave function (approximate)   Contents
G.E. Astrakharchik 15-th of December 2004